Motion for a linked group in BGE
An instance of a group is pretty much the same as an empty that adds the object in the first frame. As such, there is no relationship between a group owner and the group members unless you create one using code. Attach this to the parent of the group (I assume they are parented together to one object). Be aware that physics may not do what you expect as now the physics is associated with the empty that added the group.

import bge
cont = bge.logic.getCurrentController()
own = cont.owner

if own.groupObject is not None:
    own.setParent(own.groupObject)
else:
    print("This script isn't attached to something that is part of a group instance")